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This is a bit of a soft question, but I've wondered about it since being introduced to the $3$ isomorphism theorems (I'm aware of the $4^{th}$ as well, but it is not typically presented in the classroom as far as I know).

For setup purposes, say $G$ is a group with $H$ and $K$ normal subgroups of $G$ (and $K \subseteq H$), and $S$ a subgroup of $G$. It seems natural to ask what groups are isomorphic to $G / H$, $\frac{G / H}{H / K}$, and $SH / H$, the answers to which ultimately yield the isomorphism theorems.

I'm probably wrong here, but it seems like there must be more exotic ways of combining $G$, $H$, $K$, and $S$ (or even other (normal) subgroups) into quotient groups and asking what they are isomorphic to. My question is, why aren't there more? Of course we don't want $1729$ isomorphism theorems clogging textbooks, but why these $3$? Perhaps most / all questions of quotient groups can be determined from these $3$ theorems?

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Well, actually we only need the first isomorphism theorem, because the rest follows from this one. But for referal reasons the second and third isomorphism theorem has been made to theorems. The other theorems are lessed used and are most of the times presented as lemma's or propositions, because they are used in specific situations. This was the explanation to this question in our class. I hope this will answer your question.

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A short history lesson:

Parts of this are findable on Wikipedia. Emmy Noether generalized these theorems for modules in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern which was published in 1927. Richard Dedekind and Noether also had less general results that they had previously published.

B.L. van der Waerden published a text Algebra, the first abstract algebra textbook that took the groups-rings-fields approach to the subject, in 1930. The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when applied to groups, appeared in this text, presenting the theorems as a unified set for the first time. As van der Waerden's book was a mainstay in algebra for the better part of the next 50 (or more) years, these theorems were replicated in various forms in most every book that followed.

These theorems are useful and come up commonly. (Utility is of course in the eye of the beholder.) Also, when one theorem is presented, research and study tends to focus around it, increasing its use and application to current problems (as these problems where built around said theorem(s)).

Perhaps you could find a theorem in group theory that is simple, yet masterfully elegant, that could become the 5th isomorphism theorem!

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What about the butterfly lemma?

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  • $\begingroup$ I mentioned that I was aware of the $4^{th}$ isomorphism theorem. $\endgroup$ Commented Dec 5, 2013 at 0:54
  • $\begingroup$ Okay, I didn't know that this is the 4th isomorphism theorem. $\endgroup$ Commented Dec 5, 2013 at 0:54
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    $\begingroup$ A few select people call the Butterfly the 4th. I, for one, do not. I call the following lemma the 4th: en.wikipedia.org/wiki/Lattice_theorem $\endgroup$ Commented Dec 5, 2013 at 0:55
  • $\begingroup$ Indeed, this is a nice result. I was going off of wikipedia claiming that they are synonymous. I hope the tone didn't come across as blunt. $\endgroup$ Commented Dec 5, 2013 at 0:57

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